Continuous Wavelet Transform in Quantum Field Theory

Abstract

We describe the application of the continuous wavelet transform to calculation of the Green functions in quantum field theory: scalar $\phi^4$ theory, quantum electrodynamics, quantum chromodynamics. The method of continuous wavelet transform in quantum field theory presented in M.Altaisky Phys. Rev. D81(2010)125003 for the scalar $\phi^4$ theory, consists in substitution of the local fields $\phi(x)$ by those dependent on both the position $x$ and the resolution $a$. The substitution of the action $S[\phi(x)]$ by the action $S[\phi_a(x)]$ makes the local theory into nonlocal one, and implies the causality conditions related to the scale $a$, the region causality J.D. Christensen and L. Crane, J.Math. Phys 46 (2005) 122502. These conditions make the Green functions $G(x_1,a_1,..., x_n,a_n)=<\phi_{a_1}(x_1)...\phi_{a_n}(x_n)> $ finite for any given set of regions by means of an effective cutoff scale $A=\min (a_1,...,a_n)$.